Program
Bochner integral; Pettis and Bochner theorems; vector valued distributions and Sobolev functions.
Elements on unbounded operators: closed, dissipative and maximal dissipative operators.
Semigroups and their generators.
Cauchy problem for abstract equations, Duhamel formula.
Hille-Yosida, Lumer-Phillips and Stone theorems, construction of (semi)groups associated to Heat, Wave, Klein Gordon and Schrödinger equations.
Semilinear abstract problem, local solution, extension, global solution, continuous dependence on data.
Special properties of Heat semigroup.
Example of a nonlinear problem for Klein-Gordon equation.
Elements of interpolation theory (Three Lines and Riesz-Thorin theorem) and application to the Cauchy problem for nonlinear Schrödinger via contraction mapping theorem.
Conservation laws.
Global solutions, continuous dependence on data.